B U Department of Mathematics
Math 102 Calculus II
Fall 2005 Second Midterm
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1. Find the derivative of
f
(
x, y, z
) =
x
3
+
y
3
z
at (

1
,
2
,
1) in the direction toward (0,3,3).
Solution:
A vector in the indicated direction is (0
,
3
,
3)

(

1
,
2
,
1) =
<
1
,
1
,
2
>
.
The unit vector in that direction is
~u
=
1
√
6
<
1
,
1
,
2
>
~
O
f
(
x, y, z
) =
<
3
x
2
,
3
y
2
z, y
3
>
⇒
~
O
f
(

1
,
2
,
1) =
<
3
,
12
,
8
>
Hence, the derivative of
f
(
x, y, z
) =
x
3
+
y
3
z
at (

1
,
2
,
1) in the direction toward
(0,3,3) is